Download 6 Theory of the topological Anderson insulator

6 Theory of the topological
Anderson insulator
6.1 Introduction
Topological insulators continue to surprise with unexpected physical phenomena [30]. A recent surprise was the discovery of the
topological Anderson insulator (TAI) by Li, Chu, Jain, and Shen
[81]. In computer simulations of a HgTe quantum well, these authors discovered in the phase diagram a transition from an ordinary
insulating state (exponentially small conductance) to a state with
a quantized conductance of G0 = 2e2 /h. The name TAI refers
to the latter state. The findings of Ref. [81] were confirmed by
independent simulations [59].
The phenomenology of the TAI is similar to that of the quantum spin Hall (QSH) effect, which is well understood [62, 23]
and observed experimentally in HgTe quantum wells [70, 74, 113].
The QSH effect is a band structure effect: It requires a quantum
well with an inverted band gap, modeled by an effective Dirac
Hamiltonian with a negative (socalled “topological”) mass. The
matching of this negative mass inside the system to the usual positive mass outside leaves edge states in the gap. The edge states
are “helical”, in the sense that the direction of propagation is tied
to the electron spin. Opposite edges each contribute e2 /h to the
conductance. The conductance remains quantized in the presence
of (weak) disorder, because time reversal symmetry forbids scattering between counter-propagating edge states (of opposite helicity)
[62, 23, 70, 74, 113].
The crucial difference between the TAI and QSH phases is that
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the QSH phase extends down to zero disorder, while the TAI phase
has a boundary at a minimal disorder strength. Put differently,
the helical edge states in the QSH phase exist in spite of disorder,
while in the TAI phase they exist because of disorder. Note that the
familiar quantum Hall effect is like the QSH effect in this respect:
The edge states in the quantum Hall effect exist already without
disorder (although, unlike the QSH effect, they only form in a
strong magnetic field).
The computer simulations of Refs. [81, 59] confront us, therefore,
with a phenomenology without precedent: By what mechanism
can disorder produce edge states with a quantized conductance?
That is the question we answer in this paper.
6.2 Model
We start from the low-energy effective Hamiltonian of a HgTe
quantum well, which has the form [23]
H = α( p x σx − py σy ) + (m + βp2 )σz + [γp2 + U (r )]σ0 .
(6.1)
This is a two-dimensional Dirac Hamiltonian (with momentum
operator p = −ih̄∇, Pauli matrices σx , σy , σz , and a 2 × 2 unit
matrix σ0 ), acting on a pair of spin-orbit coupled degrees of freedom
from conduction and valence bands. The complex conjugate H ∗
acts on the opposite spin. We assume time reversal symmetry (no
magnetic field or magnetic impurities) and neglect any coupling
between the two spin blocks H and H ∗ 1 . The scalar potential U
accounts for the disorder. The parameters α, β, γ, m depend on
the thickness and composition of the quantum well [74]. For the
specific calculations that follow, we will use the same parameter
1 We
have repeated the calculations of the conductance including a coupling
Hamiltonian between the spin blocks of the form ±iκσy , with κ = 1.6 meV,
representative of bulk inversion asymmetry in a HgTe quantum well. The
effect on the phase diagram was negliglibly small.
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values as in Ref. [81], representative of a non-inverted HgTe/CdTe
quantum well2 .
The terms quadratic in momentum in Eq. (6.1) are not present
in the Dirac Hamiltonian familiar from relativistic quantum mechanics, but they play an important role here. In particular, it is the
relative sign of β and m that determines whether the clean quantum
well (U ≡ 0) is inverted (βm < 0) or not-inverted (βm > 0). We
take β > 0, so the inverted quantum well has a negative topological
mass m < 0. The inverted quantum well is a topological insulator (for Fermi energies EF inside the gap), while the non-inverted
quantum well is an ordinary band insulator. The phase transition
between these two types of insulators therefore occurs at m = 0 in
a clean quantum well.
6.3 TAI mechanism
We will now show that disorder can push the phase transition to
positive values of m, which is the hallmark of a TAI. Qualitatively,
the mechanism is as follows. Elastic scattering by a disorder potential causes states of definite momentum to decay exponentially as a
function of space and time. The quadratic term βp2 = −h̄2 β∇2 in
H, acting on the decaying state ∝ e− x/λ , adds a negative correction
δm to the topological mass. The renormalized mass m̄ = m + δm
can therefore have the opposite sign as the bare mass m. Topological mass renormalization by disorder, and the resulting change in
the phase diagram, has previously been studied without the terms
quadratic in momentum [127]. The sign of m̄ and m then remains
the same and the TAI phase cannot appear.
We extract the renormalized topological mass m̄, as well as the
renormalized chemical potential µ̄, from the self-energy Σ of the
disorder-averaged effective medium. To make contact with the
parameter values of H that we have used are: h̄α = 364.5 meV nm, h̄2 β =
686 meV nm2 , h̄2 γ = 512 meV nm2 , m = 1 meV. The lattice constant of the
discretization was a = 5 nm.
2 The
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computer simulations [81, 59], we discretize H on a square lattice
(lattice constant a) and take a random on-site disorder potential U,
uniformly distributed in the interval (−U0 /2, U0 /2). We denote
by H0 (k ) the lattice Hamiltonian of the clean quantum well in
momentum representation [23, 74].
The self-energy, defined by
( EF − H0 − Σ)−1 = h( EF − H )−1 i,
(6.2)
with h· · · i the disorder average, is a 2 × 2 matrix which we decompose into Pauli matrices: Σ = Σ0 σ0 + Σ x σx + Σy σy + Σz σz . The
renormalized topological mass and chemical potential are then
given by
m̄ = m + lim Re Σz , µ̄ = EF − lim Re Σ0 .
k →0
k →0
(6.3)
The phase boundary of the topological insulator is at m̄ = 0, while
the Fermi level enters the (negative) band gap when |µ̄| = −m̄.
In the selfconsistent Born approximation, Σ is given by the integral equation [126]
Σ=
2
2
1
12 U0 ( a/2π )
Z
BZ
dk [ EF + i0+ − H0 (k ) − Σ]−1 .
(6.4)
(The integral is over the first Brillouin zone.) The self-energy is
independent of momentum and diagonal (so there is no renormalization of the parameters α, β, γ). By calculating m̄ and µ̄ as a
function of EF and U0 we obtain the two curves A and B in Fig. 6.1.
We have also derived an approximate solution in closed form3 ,
β2 − γ2 πh̄ 4 U02 a2
β
(6.5a)
m̄ = m −
ln ,
a
48πh̄2 β2 − γ2 E2F − m2
β2 − γ2 πh̄ 4 U02 a2
γ
ln µ̄ = EF −
(6.5b)
,
a
48πh̄2 β2 − γ2 E2F − m2
3 The
approximate solution (6.5) of Eq. (6.4) amounts to the Born approximation
without selfconsistency (replacing Σ in the right-hand-side by zero) and
keeping only the logarithmically divergent part of the integral.
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showing that the correction δm = m̄ − m to the topological mass
by disorder is negative – provided β > γ. For β < γ the clean HgTe
quantum well would be a semimetal, lacking a gap in the entire
Brillouin zone. Neither the TAI phase nor the QSH phase would
then appear. In HgTe the parameter β is indeed larger than γ, but
not by much4 . Eq. (6.5) implies that the lower branch of curve B
(defined by µ̄ = m̄ < 0) is then fixed at EF ≈ m > 0 independent
of U0 . This explains the puzzling absence of the TAI phase in
the valence band (EF < 0), observed in the computer simulations
[81, 59].
To quantitatively test the phase diagram resulting from the effective medium theory, we performed computer simulations similar
to those reported in Refs. [81, 59]. The conductance G is calculated
from the lattice Hamiltonian [23, 74] in a strip geometry, using
the method of recursive Green functions. The strip consists of a
rectangular disordered region (width W, length L), connected to
semi-infinite, heavily doped, clean leads5 . Theory and simulation
are compared in Figs. 6.1 and 6.2.
Fig. 6.1 shows the phase diagram. The weak-disorder boundary
of the TAI phase observed in the simulations is described quite well
by the selfconsistent Born approximation (curve B) – without any
adjustable parameter. Curve B limits the region where (A) the renormalized topological mass m̄ is negative and (B) the renormalized
chemical potential µ̄ lies inside the band gap: |µ̄| < −m̄. Condition
(A) is needed for the existence of edge states with a quantized
conductance. Condition (B) is not needed for an infinite system,
because then Anderson localization suppresses conductance via
bulk states as well as coupling of edge states at opposite edges.
In the relatively small systems accessible by computer simulation,
the localization length for weak disorder remains larger than the
system size (see later). Condition (B) is then needed to eliminate
4 See
footnote on page 95.
reason we dope the leads in the computer simulations is to be able to access
the region | EF | < m in the phase diagram, where the band gap in the clean
leads would otherwise prevent conduction through the disordered region.
5 The
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Figure 6.1: Computer simulation of a HgTe quantum well (for parameters see footnote on page 95), showing the average
conductance h G i as a function of disorder strength U0
(logarithmic scale) and Fermi energy EF , in a disordered
strip of width W = 100 a and length L = 400 a. The
TAI phase is indicated. Curves A and B are the phase
boundaries resulting from the effective medium theory.
Curve A separates regions with positive and negative
renormalized topological mass m̄, while curve B marks
the crossing of the renormalized chemical potential µ̄
with the band edge (|µ̄| = −m̄). Both curves have been
calculated without any adjustable parameter. The phase
boundary of the TAI at strong disorder is outside of the
regime of validity of the effective medium theory.
the bulk conductance and to decouple the edge states.
Fig. 6.2 shows the average density of states ρ at the Fermi level.
The agreement between the selfconsistent Born approximation
(dashed black curve) and the computer simulation (solid black) is
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Figure 6.2: Black curves, left axis: Average density of states ρ
as a function of EF for U0 = 100 meV, calculated by
computer simulation (solid curve, for a disordered
100 a × 100 a square with periodic boundary conditions) or by effective medium theory (dashed curve).
Red curve, right axis: Average conductance h G i, calculated by computer simulation in a disordered strip
(U0 = 100 meV, W = 100 a, L = 400 a). The TAI phase
of quantized conductance lines up with the band gap.
quite good, in particular considering the fact that this plot is for a
disorder strength which is an order of magnitude larger than the
band gap. The range of Fermi energies over which the gap extends
lines up nicely with the conductance plateau, shown in the same
figure (red curve).
The strong-disorder phase boundary of the TAI cannot be described by effective medium theory, but it should be similar to the
QSH phase boundary. In the QSH effect the strong-disorder transi-
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Figure 6.3: Red dashed curve: Average conductance h G i as a function of disorder strength (EF = 25 meV, W = L = 100 a).
Black solid curve: Localization length ξ, showing the
peak at the strong-disorder edge of the conductance
plateau – characteristic of a localization transition. The
scaling with system size W of the width δU0 of the peak
is shown in the inset (double-logarithmic plot).
tion is in the universality class of the quantum Hall effect [102] –
in the absence of coupling between the spin blocks [103, 101]. To
ascertain the nature of the strong-disorder transition out of the TAI,
we have calculated the critical exponent ν governing the scaling of
the localization length ξ. For that purpose we roll up the strip into
a cylinder, thereby eliminating the edge states [59]. We determine
the localization length ξ ≡ −2 lim L→∞ Lhln G/G0 i−1 by increasing
the length L of the cylinder at fixed circumference W.
In Fig. 6.3 we show ξ as a function of disorder strength U0 at EF =
100
25 meV, W = 100 a. As mentioned above, ξ becomes much larger
than W upon crossing the weak-disorder boundary of the TAI, so
there is no localization there6 . At the strong-disorder boundary,
however, the dependence of ξ on U0 shows the characteristic peak
of a localization transition [42]. In the inset we plot the scaling
with W of the width δU0 at half maximum of the peak. This yields
the critical exponent via δU0 ∝ W −1/ν . We find ν = 2.66 ± 0.15,
consistent with the value ν = 2.59 expected for a phase transition
in the quantum Hall effect universality class [132].
6.4 Conclusion
In conclusion, we have identified the mechanism for the appearance
of a disorder-induced phase of quantized conductance in computer
simulations of a HgTe quantum well [81, 59]. The combination of
a random potential and quadratic momentum terms in the Dirac
Hamiltonian can change the sign of the topological mass, thereby
transforming a non-inverted quantum well (without edge states in
the band gap) into an inverted quantum well (with edge states).
The weak-disorder boundary in the phase diagram of the TAI has
been calculated by effective medium theory, in good agreement
with the simulations (curve B in Fig. 6.1).
Contrary to what the name “topological Anderson insulator”
might suggest, we have found that the hallmark of the TAI in the
simulations, the weak-disorder transition into a phase of quantized
conductance, is not an Anderson transition at all. Instead, the
weak-disorder boundary B marks the crossing of a band edge
rather than a mobility edge. A mobility edge (similar to the QSH
6A
pronounced conductance dip between the quantized plateau and the highconductance regime exists for samples with a large aspect ratio L/W, becoming broader and broader with increasing L/W. The dip is clearly visible in Fig.
6.2 (for L/W = 4) and absent in Fig. 6.3 (for L/W = 1). Our numerical data
suggests that the conductance dip extends over the parameter range where
W < ξ < L, so that conduction is suppressed both through the bulk and along
the edges.
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effect [102, 103, 101]) is crossed at strong disorder, as evidenced by
the localization length scaling (Fig. 6.3).
Our findings can be summed up in one sentence: “A topological insulator wants to be topological”. The mechanism for the
conversion of an ordinary insulator into a topological insulator
that we have discovered is generically applicable to narrow-band
semiconductors with strong spin-orbit coupling (since these are
described by a Dirac equation, which generically has quadratic
momentum terms [150]). There is no restriction to dimensionality.
We expect, therefore, a significant extension of the class of known
topological insulators7 to disordered materials without intrinsic
band inversion.
7 For
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introductions to topological insulators, see [149, 30, 95]